Random version of Dvoretzky's theorem in ${\mathrm{\ell }}_p^n$

We study the dependence on $\epsilon$ in the critical dimension $k(n,p,\epsilon )$ for which one can find random sections of the ${\mathrm{\ell }}_p^n$-ball which are $(1+\epsilon )$-spherical. We give lower (and upper) estimates for $k(n,p,\epsilon )$ for all eligible values $p$ and $\epsilon$ as $n\to \mathrm{\infty }$, which agree with the sharp estimates for the extreme values $p=1$ and $p=\mathrm{\infty }$. Toward this end, we provide tight bounds for the Gaussian concentration of the ${\mathrm{\ell }}_p$-norm.